2.3 Model Processes

Mathematical formulations for each model process are provided in the following sections and computer algorithms for the binomial and hypergeometric distributions are provided in Appendices A and B. As noted in Section 2.0, all model processes will have the capability to be run in either a deterministic or stochastic mode. Notation used in the formulas is provided in Appendix C.

2.3.1 Stochastic Variation of Abundance

At the initiation of each Monte Carlo repetition, the survival of fish from smolt to recruitment at January of age 3 is modelled using a normal distribution.

2.3.2 Natural Mortality

At the initiation of each time step, the number of fish which survive from natural mortality is computed using a binomial distribution:

[1]
[2]

2.3.3 Dispersion

In the initial time step, stocks are dispersed to fishing areas using a multinomial distribution:

[3]

For subsequent time intervals, the number of fish from a stock which are in area a' is the sum of fish which either remain in the area or emigrate into it:

[4]

where

[5]

2.3.4 Fishing Mortality

Fishery mortality in the SFM can be controlled using seven different mechanisms:

1. Catch Quota - The target harvest in a fishery in a time step is set equal to an input quota;

2. Escapement Goal - The target annual catch in a fishery harvesting mature fish is constrained to allow a specified number to pass to escapement. Escapement will occur in the model as a result of either: (1) dispersion of fish from a fishing area; or (2) fish escaping from a terminal harvest.

3. Fishing Effort - The harvest rate in a fishery in a time step is a function of the input effort and a catchability coefficient;

4. Catch Ceiling - The target harvest in a fishery in a time step is set equal to the catch ceiling unless harvesting the ceiling would require an increase in the harvest rate from the base period level.

5. Seasonal Catch Ceiling - The target harvest in a fishery across a specified number of time steps is set equal to the catch ceiling unless harvesting the ceiling would require an increase in the harvest rate from the base period level.

6. Selective Fishery - Fish encountered which are marked for selective removal may be retained. The appropriate probability distribution to model the number of marked fish captured in a selective fishery depends upon the fate of the unmarked fish which are released. If unmarked fish are susceptible to recapture, the correct distribution is a generalized Polya distribution (Johnson and Kotz, 1977). Conversely, if released fish are not susceptible to recapture within the time period, the appropriate distribution is a hypergeometric. Analysis indicates that the two options do not differ appreciably when harvest rates within a time period are less than 30% (Fig. 2-1). The multiple hypergeometric distribution is used in the SFM for the following reasons: (1) Harvest rates are likely to be less than 30% within the weekly time step of the model; (2) some reduced susceptibility to recapture within a week is likely; and (3) the probabilities are much more easily computed.

7. Daily Bag Limit - The catch of individual fishermen is limited to the daily bag limit. The bag limit may be for total marked and unmarked fish (traditional bag limit), marked fish (selective fishery with bag limit), or for a total number of fish of which a specified number may be unmarked (selective fishery with bag limit of marked and unmarked fish); or

8. Size Limit - All fish of length greater than the size limit may be retained (this type of fishery control may be used in conjunction with types 1-7).

For each of the harvest controls listed above, the total encounters are modified to account for dropoff mortality, which is estimated by multiplying the total encounters by time step by the binomial estimate of a fixed fraction.

2.3.4.1 Computation of Fishing Mortalities Associated With Catch Quotas

The target harvest in a fishery in a time step is set equal to an input quota. Catch quotas are simulated using a compound gamma-multiple hypergeometric distribution (the multiple hypergeometric is approximated through iterative application of an univariate hypergeometric). The gamma distribution will be used to simulate management error in achieving the catch quota.

[6]

Conditional upon C*, the catch by stock in the fishery will be given by:

[7]

2.3.4.2 Computation of Fishing Mortalities Associated With Escapement Goals

The escapement goal fishery control algorithm will be applicable only to fishing areas immediately adjacent to the spawning grounds (terminal areas) and only to one gear type. Once fish enter these terminal fishing areas, they are either harvested within the area or pass to escapement. Placing these constraints upon the terminal area fisheries obviates the need for a iterative loop to implement an escapement goal fishery control.

Let the harvest in terminal fishing area a´ be controlled by stock s´. Then the terminal run for the stock will be

[8]

The allowable harvest rate in the area will be

[9]

The target catch in the fishery of the S* stocks present will be

[10]

As with the catch quota, the gamma distribution will be used to simulate management error in achieving the target catch.

[11]

The catch will be allocated to stocks using a multiple hypergeometric distribution.

[12]

2.3.4.3 Computation of Fishing Mortalities With All Other Catch Constraints

The steps below outline the model protocol for computing mortalities by category. This protocol is common to all model fisheries except those that are controlled by quotas or escapement goals.

Step 1. The total encounters in the fishery are obtained by:

[13]

where

[14]

and error in the expected effort is simulated using a gamma distribution.

[15]

Step 2. The total encounters are then allocated to the four categories: (1) marked and tagged (MT); (2) marked and untagged (MU); (3) unmarked and tagged (UT); and (4) unmarked and untagged (UU) using a multiple hypergeometric distribution.

INSERT EQUATION

Step 3. Compute the dropoff mortality for each of the four categories from Step 2.

INSERT EQUATION

Step 4. Compute the size limit effect for each category. All fish of length greater than the size limit may be retained. For fisheries with a minimum size limit, the probability that a fish is less than the minimum size limit will be computed from a normal distribution:

[16]

The number of sublegal sized fish encountered in an effort fishery will be computed by

[17]

Mortality of fish less than the size limit will be computed using a binomial equation.

[18]

Step 5. If the fishery is selective, compute steps 5a-d, if the fishery is not selective go to Step 6.

Step 5a. Compute the numbers of UT and UU fish that are released using a binomial distribution.

[19]

Step 5b. Compute the mortality of UT and UU fish released using a binomial distribution.

[20]

Step 5c. Compute the numbers of MT and MU fish that are retained or released using a binomial distribution.

[21]

Step 5d. Compute the mortality of MT and MU fish released using an equation similar to 16.

Step 6. Compute the effects of a bag limit if applicable. The catch of individual fishermen is limited to the daily bag limit. Three types of daily bag limits per unit of effort may be specified. Catch may be controlled by: (1) A limitation on the total number of fish retained (traditional bag limit); (2) a limitation on the number of unmarked fish retained (selective fishery); or (3) a limitation on the number of marked and unmarked fish which may be retained.

Case 1 - Traditional Bag Limit. The method of Porch and Fox (1991) will be used with slight modification:

Step 6-1a. Compute the number of fish encountered without a bag limit using equation 10;

Step 6-1b. Generate the frequency distribution of encounters per unit of effort using a negative binomial distribution;

Step 6-1c. Censor the distribution generated in Step 2 to obtain the catch with the bag limit;

Step 6-1d. Allocate the catch to stock using a multiple hypergeometric distribution (equation 13).

Case 2 - Selective Fishery With Bag Limit on Marked Fish. Case 1 will be generalized for a bag limit on marked fish as follows.

Step 6-2a. Compute the number of marked fish encountered using a binomial distribution similar to equation 10.

[22]

Step 6-2b. Compute the number of marked fish which would be identified and retained in the absence of a bag limit using a binomial distribution conditional on the number of marked fish encountered.

[23]

Step 6-2c. Generate the theoretical frequency distribution of encounters per unit of effort in the absence of a bag limit using a negative binomial distribution.

Step 6-2d. Censor the distribution generated in Step 3 to obtain the catch with the bag limit of marked fish.

Step 6-2e. Compute the actual effort in the fishery by solving for the effort in the equation

[24]

where c is a set equal to

[25]

Step 6-2f. Use a negative binomial distribution conditioned on the catch per unit of effort values obtained in Step 6-2d to compute the number of unmarked fish encountered before the catch per unit effort value is obtained (C/f successes).

[26]

Step 6-2g. Compute the total number of unmarked fish encountered by multiplying the effort by the product of the frequency distribution of catch per unit effort (Step 4) and the distribution of the number of unmarked fish encountered (Step 5).

[27]

The remainder of the computations will be similar to Steps 5a-d.

Case 3 - Selective Fishery With Bag Limit on Marked and Unmarked Fish. Computational procedures will be identical to Case 2 except that the lower limit of the range of the summation in equation 26 will be the number of unmarked fish which may be retained plus 1.

[28]

Step 7. Compare catch with seasonal ceiling if applicable, and adjust catches if necessary.

INSERT EQUATION.

Step 8. Allocate mortalities in all categories to stock using multiple hypergeometric distributions:

[29]